Galois action on knots II: Proalgebraic string links and knots

TL;DR

This paper explores the action of the Grothendieck-Teichmüller proalgebraic group on proalgebraic tangles, revealing a motivic structure and providing explicit calculations for knots and string links.

Contribution

It introduces a new motivic framework for proalgebraic tangles and computes the inverse image of trivial chord diagrams under the Kontsevich isomorphism.

Findings

Identifies unique properties of the group action on proalgebraic knots and string links.

Explicitly calculates the inverse image of the trivial chord diagram.

Reveals a motivic structure on tangles.

Abstract

We discuss an action of the Grothendieck-Teichm\"{u}ller proalgebraic group on the linear span of proalgebraic tangles, oriented tangles completed by a filtration of Vassiliev. The action yields a motivic structure on tangles. We derive distinguished properties of the action particularly on proalgebraic string links and on proalgebraic knots which can not be observed in the action on proalgebraic braids. By exploiting the properties, we explicitly calculate the inverse image of the trivial (the chordless) chord diagram under the Kontsevich isomorphism.